Question

Simple Linear The following information regarding a dependent variable (Y) and an independent variable (X) is provided.

Y 4,5,9,12,14

X 8,5,3,2,1

a. For the above observations, plot a scatter diagram and indicate what kind of relationship (if any) exist between x and y

. b. Find the estimate simple linear relationship between x and y

c. Find MSE

d. Find the coefficient of determination

e. Find and interpret the correlation coefficient

f. Use the F statistic to test for any significant relationship between x and y at 95% confidence. Comment on your finding

g. Use the T statistic to test for any significant relationship between x and y at 95% confidence. Comment on your finding

Answer 1

a)

Steps for Scatterplot in Excel

• Import the data in Excel
• In Insert Tab Click on scatterplot and select the Y values and X values. Click OK

From the above scatterplot, we can observer an inverse relationship between x and y ie as x increases, y decreases and vice versa.

b)

Steps for Regression in Excel

• Import the data in Excel
• In Data Analysis tool from Data Tab.
• Click on regression and select the Y values and X values. Select Labels and click OK.

Regression Output

Regression Statistics
Multiple R	0.9417
R Square	0.8868
Adjusted R Square	0.8491
Standard Error	1.6800
Observations	5
ANOVA
	df	SS	MS	F	Significance F
Regression	1	66.33	66.33	23.50	0.02
Residual	3	8.47	2.82
Total	4	74.80
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	14.38	1.37	10.46	0.00	10.00	18.75
X	-1.47	0.30	-4.85	0.02	-2.43	-0.50

Regression Equation

Y = 14.38 – 1.47 X

C)

From the Output

MSE = 2.82

d)

Coefficient of determination = R Square = 0.8868 ie 88.68%

R Square tell us that 88.68% of the variation in y can be explained by x

e)

Correlation coefficient = -0.9417

Magnitude of Correlation coefficient indicates that there is strong correlation between x and y.

Negative sign indicates inverse relationship ie as x increases, y decreases and vice versa.

f)

alpha = 0.05

Null and Alternate Hypothesis

H₀: All the coefficients are zero (ie the model is irrelevant)

H_a: Not all the coefficients are zero (ie the model is relevant)

Test Statistic

F = 23.50

p-value = 0.02

Result

Since the p-value is less than 0.05, we reject the null hypothesis in favour of alternate hypothesis

Conclusion

There is significant relationship between x and y

g)

alpha = 0.05

Null and Alternate Hypothesis

H₀: coefficient of x is zero (ie x is not significant)

H_a: coefficient of x is not zero (ie x is significant)

Test Statistic

t = -4.85

p-value = 0.02

Result

Since the p-value is less than 0.05, we reject the null hypothesis in favour of alternate hypothesis

Conclusion

There is significant relationship between x and y

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